Controllability Of A One-dimensional Wave Equation In Non-cylindrical Domains

This dissertation investigates mainly some controllability problems of a one-dimensional wave equation in non-cylindrical domains. This equation characterizes the motion of a finite string. We put different controls on this system. Then we obtain boundary controllability and locally interior controllability.This dissertation consists of three parts.In the first part of this dissertation, i.e. Chapter2and Chapter3, we study boundary controllability of a one-dimensional wave equation with a fixed endpoint and the other moving one. We put controls on the moving endpoint. In Chapter2, when speed of the moving end point is less than the characteristic speed of the wave equation, we derive partial Dirichlet boundary controllability and Dirichlet-Neumann boundary controllability by Hilbert Uniqueness Method. In Chapter3, assume that speed of the moving end point be equal with the characteristic speed of the wave equation, we shall prove existence and uniqueness of classical solution for the wave equation of Cauchy-Goursat type. Moreover, we give expression of the system by constructive method. And by the specific expression of the solu-tion, sufficient and necessary conditions which ensure the exact controllability are formulated.In the second part of this dissertation, i.e. Chapter4, we are concerned with boundary controllability of a one-dimensional wave equation with a fixed endpoint and the other moving one. We put controls on the fixed endpoint. By Hilbert Uniqueness Method, we get partial Dirichlet boundary controllability. When we prove observation inequality, we use sidewise energy estimates method and multi-plier method to obtain it.In the third part of this dissertation, i.e. Chapter5, we consider local dis-tributed control of this system. We prove exact internal controllability, employing operator theory.